2
$\begingroup$

Let $\pi: X \rightarrow S$ be the Neron model of an elliptic curve over a dedekind domain (but probably any minimal elliptic fibration will suffice).

Let $\eta$ be the generic point of $S$, $K = S(\eta)$ and $ 0_\eta \in X(K)$ denote the zero element of the generic fiber. Suppose $\alpha$ is a section of $\pi$ and specifies a torsion element of the generic fiber $E(K)$. Suppose that $X$ has a fiber of Kodaira type $I_n$ with $n> 1$ at $p \in S$. There is a (nameless?) natural specialization homomorphism $$ E(K)_{tors} \xrightarrow{\phi} G_p $$ to the group of irreducible components of the fiber over $p$: elements of $E(K)$ correspond to sections of $\pi$, and such a section meets exactly one irreducible component of an $I_n$ fiber. Label the components $0, 1, ..., n-1$ starting with the component the $0$-element meets and then going around the $I_n$ fiber consecutively (either direction).

$\textbf{Question:}$ For my own sake, I would like to know how to calculate the image of a torsion point under this map - the completely naive approach (represent torsion points as sections, and calculate intersections in the Neron Severi group) is not very computable - either by hand or computer.

Dispite it's simplicity, I can't find any resources that might explain how to do this. In particular, I didn't find anything useful in BLR's Neron Models book. For generalities on this map where $S$ is a smooth proper curve over $\mathbb{C}$,see p.75 of Miranda's Book http://www.math.colostate.edu/~miranda/BTES-Miranda.pdf

$\endgroup$
1

1 Answer 1

2
$\begingroup$

Your question is local, so we may take $S$ to be the ring of integers in a local field $K$. If the point $P\in E(K)$ is torsion or not does not matter in my answer here. I think here of an ellpitic curve over a number field.

I think the best theoretical tool to study fibres of type $I_n$ is by using Tate's uniformisation. This works over $K$ if the reduction is split multiplicative. If it is non-split then the question is rather easier as the group of components is small.

Tate's parametrisation gives an isomorphism of $E(K)$ to $K^{\times}/q^{\mathbb{Z}}$ for some $q$ in $K$ of valuation $n$. The components can now simply be labeled by the valuation map $E(K) \to K^{\times}/q^{\mathbb{Z}} \to \mathbb{Z}/n\mathbb{Z}$. See Silverman 2.

Alternatively, for an explicit approach one can use Tate's algorithm. More precisely, Tate's algorithm is a simplification of the concrete blow-ups needed to build the minimal regular model. One can now follow the point $P\in E(K)$ along these blow-ups and find the corresponding label of the component. The function component_of_a_point(P,v) in https://www.maths.nottingham.ac.uk/personal/cw/download/class_group_pairing.py does exactly what you want.

Finally, not that the index is not exactly well-defined. It is really in $\mathbb{Z}/n\mathbb{Z}$ modulo $\pm 1$ as we can label in one direction or the other.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.